Necking limit of substrate-supported metal layers under biaxial in-plane loading

Necking instability often indicates the onset of ductile failure. It has been shown that the necking instability in a substrate-supported metal layer can be retarded to a higher strain than that in a single freestanding metal layer. Most existing theoretical studies of the necking limit of substrate-supported metal layers assume plane strain condition. However, most commonly conducted experiments of such metal/substrate bilayers are uniaxial tensile tests. So far, the necking instability of substrate-supported metal layers under arbitrary combinations of biaxial in-plane loading conditions remains poorly understood. This paper presents a comprehensive study of the necking limit of a metal/substrate bilayer over the full range of biaxial loading ratio, from 1 for equibiaxial loading, to 0 for plane strain loading, and to −1/2 for uniaxial loading. Two representative material combinations are considered, namely, a metal layer supported by a stiff plastic substrate, and a metal layer supported by a compliant elastomer substrate. The results quantitatively correlate both critical necking limit strain and necking band orientation with the material properties and thickness ratio of the substrate-metal bilayer. In particular, the predicted necking band orientation when the bilayer is under in-plane loading with a negative ratio (e.g., uniaxial tension) agrees with the slanted necking bands observed in experiments, a phenomenon that cannot be explained by existing theoretical studies assuming plane strain condition. The present study further shows that necking retardation in an elastomer-supported metal layer can allow the bilayer to absorb and dissipate more energy than an all-metal single layer with the same mass. These understandings shed light on optimal design of substrate-supported metal structures with enhanced deformability and energy absorbing capacity under complex in-plane loading conditions.